LMFDB - Embedded newform 4304.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$34.3676130300$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.14366$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.65255 q^{3} +0.725043 q^{5} -1.29978 q^{7} -0.269065 q^{9} +3.71687 q^{11} +2.30856 q^{13} -1.19817 q^{15} +6.32982 q^{17} -0.0212539 q^{19} +2.14795 q^{21} +1.30511 q^{23} -4.47431 q^{25} +5.40231 q^{27} -2.06303 q^{29} +5.82316 q^{31} -6.14233 q^{33} -0.942394 q^{35} -7.25272 q^{37} -3.81503 q^{39} +8.77454 q^{41} -4.52327 q^{43} -0.195084 q^{45} -3.85502 q^{47} -5.31058 q^{49} -10.4604 q^{51} -6.86923 q^{53} +2.69489 q^{55} +0.0351232 q^{57} -7.84518 q^{59} +3.84958 q^{61} +0.349724 q^{63} +1.67381 q^{65} +9.31541 q^{67} -2.15676 q^{69} +0.643097 q^{71} -2.97920 q^{73} +7.39404 q^{75} -4.83110 q^{77} -0.418245 q^{79} -8.12041 q^{81} +15.6322 q^{83} +4.58939 q^{85} +3.40927 q^{87} +10.3879 q^{89} -3.00062 q^{91} -9.62309 q^{93} -0.0154100 q^{95} +5.07659 q^{97} -1.00008 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} + 7 q^{5} - 6 q^{7} + 12 q^{9} + 3 q^{11} - 9 q^{13} - 8 q^{15} + 8 q^{17} + 11 q^{19} - 6 q^{21} - 12 q^{23} + 22 q^{25} + 14 q^{27} - 5 q^{29} - 14 q^{31} - 4 q^{33} + 4 q^{35} + 13 q^{37}+ \cdots + 37 q^{99}+O(q^{100}) $ 7 * q - q^3 + 7 * q^5 - 6 * q^7 + 12 * q^9 + 3 * q^11 - 9 * q^13 - 8 * q^15 + 8 * q^17 + 11 * q^19 - 6 * q^21 - 12 * q^23 + 22 * q^25 + 14 * q^27 - 5 * q^29 - 14 * q^31 - 4 * q^33 + 4 * q^35 + 13 * q^37 + 18 * q^39 + 12 * q^41 + 11 * q^43 + 3 * q^45 - 2 * q^47 + 15 * q^49 + 26 * q^51 + 19 * q^53 + 40 * q^55 - 12 * q^57 + 9 * q^59 - 3 * q^61 + 26 * q^63 - 10 * q^65 + 33 * q^67 - 64 * q^69 - 28 * q^71 - 14 * q^73 + 45 * q^75 + 10 * q^77 - 2 * q^79 + 15 * q^81 + 7 * q^83 - 16 * q^85 - 16 * q^87 + 18 * q^89 + 26 * q^91 + 6 * q^93 + 34 * q^95 - 4 * q^97 + 37 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.65255	−0.954103	−0.477051	−	0.878875i	$-0.658294\pi$
$3$	−1.65255	−0.954103	−0.477051	+	0.878875i	$0.658294\pi$
$4$	0	0
$5$	0.725043	0.324249	0.162125	−	0.986770i	$-0.448165\pi$
$5$	0.725043	0.324249	0.162125	+	0.986770i	$0.448165\pi$
$6$	0	0
$7$	−1.29978	−0.491269	−0.245635	−	0.969362i	$-0.578996\pi$
$7$	−1.29978	−0.491269	−0.245635	+	0.969362i	$0.578996\pi$
$8$	0	0
$9$	−0.269065	−0.0896884
$10$	0	0
$11$	3.71687	1.12068	0.560340	−	0.828263i	$-0.310671\pi$
$11$	3.71687	1.12068	0.560340	+	0.828263i	$0.310671\pi$
$12$	0	0
$13$	2.30856	0.640280	0.320140	−	0.947370i	$-0.396270\pi$
$13$	2.30856	0.640280	0.320140	+	0.947370i	$0.396270\pi$
$14$	0	0
$15$	−1.19817	−0.309367
$16$	0	0
$17$	6.32982	1.53521	0.767603	−	0.640925i	$-0.221449\pi$
$17$	6.32982	1.53521	0.767603	+	0.640925i	$0.221449\pi$
$18$	0	0
$19$	−0.0212539	−0.00487598	−0.00243799	−	0.999997i	$-0.500776\pi$
$19$	−0.0212539	−0.00487598	−0.00243799	+	0.999997i	$0.500776\pi$
$20$	0	0
$21$	2.14795	0.468721
$22$	0	0
$23$	1.30511	0.272134	0.136067	−	0.990700i	$-0.456554\pi$
$23$	1.30511	0.272134	0.136067	+	0.990700i	$0.456554\pi$
$24$	0	0
$25$	−4.47431	−0.894863
$26$	0	0
$27$	5.40231	1.03967
$28$	0	0
$29$	−2.06303	−0.383095	−0.191548	−	0.981483i	$-0.561351\pi$
$29$	−2.06303	−0.383095	−0.191548	+	0.981483i	$0.561351\pi$
$30$	0	0
$31$	5.82316	1.04587	0.522935	−	0.852372i	$-0.324837\pi$
$31$	5.82316	1.04587	0.522935	+	0.852372i	$0.324837\pi$
$32$	0	0
$33$	−6.14233	−1.06924
$34$	0	0
$35$	−0.942394	−0.159294
$36$	0	0
$37$	−7.25272	−1.19234	−0.596170	−	0.802858i	$-0.703312\pi$
$37$	−7.25272	−1.19234	−0.596170	+	0.802858i	$0.703312\pi$
$38$	0	0
$39$	−3.81503	−0.610893
$40$	0	0
$41$	8.77454	1.37035	0.685176	−	0.728377i	$-0.259725\pi$
$41$	8.77454	1.37035	0.685176	+	0.728377i	$0.259725\pi$
$42$	0	0
$43$	−4.52327	−0.689792	−0.344896	−	0.938641i	$-0.612086\pi$
$43$	−4.52327	−0.689792	−0.344896	+	0.938641i	$0.612086\pi$
$44$	0	0
$45$	−0.195084	−0.0290814
$46$	0	0
$47$	−3.85502	−0.562313	−0.281156	−	0.959662i	$-0.590718\pi$
$47$	−3.85502	−0.562313	−0.281156	+	0.959662i	$0.590718\pi$
$48$	0	0
$49$	−5.31058	−0.758654
$50$	0	0
$51$	−10.4604	−1.46474
$52$	0	0
$53$	−6.86923	−0.943562	−0.471781	−	0.881716i	$-0.656389\pi$
$53$	−6.86923	−0.943562	−0.471781	+	0.881716i	$0.656389\pi$
$54$	0	0
$55$	2.69489	0.363379
$56$	0	0
$57$	0.0351232	0.00465218
$58$	0	0
$59$	−7.84518	−1.02136	−0.510678	−	0.859772i	$-0.670605\pi$
$59$	−7.84518	−1.02136	−0.510678	+	0.859772i	$0.670605\pi$
$60$	0	0
$61$	3.84958	0.492888	0.246444	−	0.969157i	$-0.420738\pi$
$61$	3.84958	0.492888	0.246444	+	0.969157i	$0.420738\pi$
$62$	0	0
$63$	0.349724	0.0440611
$64$	0	0
$65$	1.67381	0.207610
$66$	0	0
$67$	9.31541	1.13806	0.569029	−	0.822317i	$-0.307319\pi$
$67$	9.31541	1.13806	0.569029	+	0.822317i	$0.307319\pi$
$68$	0	0
$69$	−2.15676	−0.259644
$70$	0	0
$71$	0.643097	0.0763215	0.0381608	−	0.999272i	$-0.487850\pi$
$71$	0.643097	0.0763215	0.0381608	+	0.999272i	$0.487850\pi$
$72$	0	0
$73$	−2.97920	−0.348689	−0.174345	−	0.984685i	$-0.555781\pi$
$73$	−2.97920	−0.348689	−0.174345	+	0.984685i	$0.555781\pi$
$74$	0	0
$75$	7.39404	0.853791
$76$	0	0
$77$	−4.83110	−0.550555
$78$	0	0
$79$	−0.418245	−0.0470563	−0.0235281	−	0.999723i	$-0.507490\pi$
$79$	−0.418245	−0.0470563	−0.0235281	+	0.999723i	$0.507490\pi$
$80$	0	0
$81$	−8.12041	−0.902268
$82$	0	0
$83$	15.6322	1.71585	0.857927	−	0.513771i	$-0.171752\pi$
$83$	15.6322	1.71585	0.857927	+	0.513771i	$0.171752\pi$
$84$	0	0
$85$	4.58939	0.497789
$86$	0	0
$87$	3.40927	0.365512
$88$	0	0
$89$	10.3879	1.10111	0.550556	−	0.834798i	$-0.314416\pi$
$89$	10.3879	1.10111	0.550556	+	0.834798i	$0.314416\pi$
$90$	0	0
$91$	−3.00062	−0.314550
$92$	0	0
$93$	−9.62309	−0.997868
$94$	0	0
$95$	−0.0154100	−0.00158103
$96$	0	0
$97$	5.07659	0.515450	0.257725	−	0.966218i	$-0.417027\pi$
$97$	5.07659	0.515450	0.257725	+	0.966218i	$0.417027\pi$
$98$	0	0
$99$	−1.00008	−0.100512
$100$	0	0
$101$	12.5321	1.24699	0.623495	−	0.781827i	$-0.285712\pi$
$101$	12.5321	1.24699	0.623495	+	0.781827i	$0.285712\pi$
$102$	0	0
$103$	−5.43374	−0.535403	−0.267701	−	0.963502i	$-0.586264\pi$
$103$	−5.43374	−0.535403	−0.267701	+	0.963502i	$0.586264\pi$
$104$	0	0
$105$	1.55736	0.151982
$106$	0	0
$107$	3.45500	0.334007	0.167004	−	0.985956i	$-0.446591\pi$
$107$	3.45500	0.334007	0.167004	+	0.985956i	$0.446591\pi$
$108$	0	0
$109$	−6.80801	−0.652089	−0.326044	−	0.945354i	$-0.605716\pi$
$109$	−6.80801	−0.652089	−0.326044	+	0.945354i	$0.605716\pi$
$110$	0	0
$111$	11.9855	1.13762
$112$	0	0
$113$	0.407582	0.0383421	0.0191710	−	0.999816i	$-0.493897\pi$
$113$	0.407582	0.0383421	0.0191710	+	0.999816i	$0.493897\pi$
$114$	0	0
$115$	0.946259	0.0882391
$116$	0	0
$117$	−0.621154	−0.0574257
$118$	0	0
$119$	−8.22735	−0.754200
$120$	0	0
$121$	2.81514	0.255922
$122$	0	0
$123$	−14.5004	−1.30746
$124$	0	0
$125$	−6.86928	−0.614407
$126$	0	0
$127$	9.52589	0.845286	0.422643	−	0.906296i	$-0.361102\pi$
$127$	9.52589	0.845286	0.422643	+	0.906296i	$0.361102\pi$
$128$	0	0
$129$	7.47495	0.658132
$130$	0	0
$131$	21.9325	1.91625	0.958127	−	0.286342i	$-0.0924394\pi$
$131$	21.9325	1.91625	0.958127	+	0.286342i	$0.0924394\pi$
$132$	0	0
$133$	0.0276253	0.00239542
$134$	0	0
$135$	3.91690	0.337113
$136$	0	0
$137$	−2.48849	−0.212606	−0.106303	−	0.994334i	$-0.533901\pi$
$137$	−2.48849	−0.212606	−0.106303	+	0.994334i	$0.533901\pi$
$138$	0	0
$139$	−21.6244	−1.83416	−0.917078	−	0.398708i	$-0.869459\pi$
$139$	−21.6244	−1.83416	−0.917078	+	0.398708i	$0.869459\pi$
$140$	0	0
$141$	6.37063	0.536504
$142$	0	0
$143$	8.58064	0.717549
$144$	0	0
$145$	−1.49579	−0.124218
$146$	0	0
$147$	8.77602	0.723834
$148$	0	0
$149$	20.9106	1.71306	0.856532	−	0.516093i	$-0.172614\pi$
$149$	20.9106	1.71306	0.856532	+	0.516093i	$0.172614\pi$
$150$	0	0
$151$	2.07209	0.168624	0.0843120	−	0.996439i	$-0.473131\pi$
$151$	2.07209	0.168624	0.0843120	+	0.996439i	$0.473131\pi$
$152$	0	0
$153$	−1.70313	−0.137690
$154$	0	0
$155$	4.22204	0.339123
$156$	0	0
$157$	−2.60975	−0.208281	−0.104140	−	0.994563i	$-0.533209\pi$
$157$	−2.60975	−0.208281	−0.104140	+	0.994563i	$0.533209\pi$
$158$	0	0
$159$	11.3518	0.900255
$160$	0	0
$161$	−1.69635	−0.133691
$162$	0	0
$163$	6.63147	0.519417	0.259708	−	0.965687i	$-0.416374\pi$
$163$	6.63147	0.519417	0.259708	+	0.965687i	$0.416374\pi$
$164$	0	0
$165$	−4.45345	−0.346701
$166$	0	0
$167$	−22.6369	−1.75170	−0.875850	−	0.482584i	$-0.839698\pi$
$167$	−22.6369	−1.75170	−0.875850	+	0.482584i	$0.839698\pi$
$168$	0	0
$169$	−7.67053	−0.590041
$170$	0	0
$171$	0.00571868	0.000437318 0
$172$	0	0
$173$	12.2456	0.931018	0.465509	−	0.885043i	$-0.345871\pi$
$173$	12.2456	0.931018	0.465509	+	0.885043i	$0.345871\pi$
$174$	0	0
$175$	5.81561	0.439619
$176$	0	0
$177$	12.9646	0.974478
$178$	0	0
$179$	18.0181	1.34673	0.673367	−	0.739308i	$-0.264847\pi$
$179$	18.0181	1.34673	0.673367	+	0.739308i	$0.264847\pi$
$180$	0	0
$181$	7.77690	0.578052	0.289026	−	0.957321i	$-0.406669\pi$
$181$	7.77690	0.578052	0.289026	+	0.957321i	$0.406669\pi$
$182$	0	0
$183$	−6.36163	−0.470266
$184$	0	0
$185$	−5.25854	−0.386615
$186$	0	0
$187$	23.5271	1.72047
$188$	0	0
$189$	−7.02179	−0.510760
$190$	0	0
$191$	−0.312243	−0.0225931	−0.0112965	−	0.999936i	$-0.503596\pi$
$191$	−0.312243	−0.0225931	−0.0112965	+	0.999936i	$0.503596\pi$
$192$	0	0
$193$	−0.336949	−0.0242541	−0.0121271	−	0.999926i	$-0.503860\pi$
$193$	−0.336949	−0.0242541	−0.0121271	+	0.999926i	$0.503860\pi$
$194$	0	0
$195$	−2.76606	−0.198082
$196$	0	0
$197$	26.8275	1.91138	0.955691	−	0.294370i	$-0.0951099\pi$
$197$	26.8275	1.91138	0.955691	+	0.294370i	$0.0951099\pi$
$198$	0	0
$199$	−4.03560	−0.286076	−0.143038	−	0.989717i	$-0.545687\pi$
$199$	−4.03560	−0.286076	−0.143038	+	0.989717i	$0.545687\pi$
$200$	0	0
$201$	−15.3942	−1.08582
$202$	0	0
$203$	2.68148	0.188203
$204$	0	0
$205$	6.36192	0.444335
$206$	0	0
$207$	−0.351159	−0.0244072
$208$	0	0
$209$	−0.0789980	−0.00546440
$210$	0	0
$211$	−9.22688	−0.635205	−0.317602	−	0.948224i	$-0.602878\pi$
$211$	−9.22688	−0.635205	−0.317602	+	0.948224i	$0.602878\pi$
$212$	0	0
$213$	−1.06275	−0.0728186
$214$	0	0
$215$	−3.27956	−0.223664
$216$	0	0
$217$	−7.56881	−0.513804
$218$	0	0
$219$	4.92329	0.332685
$220$	0	0
$221$	14.6128	0.982963
$222$	0	0
$223$	9.44842	0.632713	0.316357	−	0.948640i	$-0.397540\pi$
$223$	9.44842	0.632713	0.316357	+	0.948640i	$0.397540\pi$
$224$	0	0
$225$	1.20388	0.0802588
$226$	0	0
$227$	−15.2061	−1.00926	−0.504632	−	0.863335i	$-0.668372\pi$
$227$	−15.2061	−1.00926	−0.504632	+	0.863335i	$0.668372\pi$
$228$	0	0
$229$	28.6411	1.89266	0.946329	−	0.323205i	$-0.104760\pi$
$229$	28.6411	1.89266	0.946329	+	0.323205i	$0.104760\pi$
$230$	0	0
$231$	7.98366	0.525286
$232$	0	0
$233$	3.14875	0.206281	0.103141	−	0.994667i	$-0.467111\pi$
$233$	3.14875	0.206281	0.103141	+	0.994667i	$0.467111\pi$
$234$	0	0
$235$	−2.79506	−0.182329
$236$	0	0
$237$	0.691173	0.0448965
$238$	0	0
$239$	−12.5059	−0.808940	−0.404470	−	0.914551i	$-0.632544\pi$
$239$	−12.5059	−0.808940	−0.404470	+	0.914551i	$0.632544\pi$
$240$	0	0
$241$	3.73027	0.240287	0.120144	−	0.992757i	$-0.461664\pi$
$241$	3.73027	0.240287	0.120144	+	0.992757i	$0.461664\pi$
$242$	0	0
$243$	−2.78751	−0.178819
$244$	0	0
$245$	−3.85040	−0.245993
$246$	0	0
$247$	−0.0490660	−0.00312199
$248$	0	0
$249$	−25.8330	−1.63710
$250$	0	0
$251$	26.2795	1.65875	0.829374	−	0.558694i	$-0.188697\pi$
$251$	26.2795	1.65875	0.829374	+	0.558694i	$0.188697\pi$
$252$	0	0
$253$	4.85092	0.304975
$254$	0	0
$255$	−7.58421	−0.474942
$256$	0	0
$257$	28.2237	1.76055	0.880273	−	0.474468i	$-0.157359\pi$
$257$	28.2237	1.76055	0.880273	+	0.474468i	$0.157359\pi$
$258$	0	0
$259$	9.42692	0.585760
$260$	0	0
$261$	0.555090	0.0343592
$262$	0	0
$263$	−25.3865	−1.56540	−0.782701	−	0.622398i	$-0.786158\pi$
$263$	−25.3865	−1.56540	−0.782701	+	0.622398i	$0.786158\pi$
$264$	0	0
$265$	−4.98049	−0.305949
$266$	0	0
$267$	−17.1665	−1.05057
$268$	0	0
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	17.9078	1.08782	0.543911	−	0.839143i	$-0.316943\pi$
$271$	17.9078	1.08782	0.543911	+	0.839143i	$0.316943\pi$
$272$	0	0
$273$	4.95868	0.300113
$274$	0	0
$275$	−16.6304	−1.00285
$276$	0	0
$277$	−6.51988	−0.391741	−0.195871	−	0.980630i	$-0.562753\pi$
$277$	−6.51988	−0.391741	−0.195871	+	0.980630i	$0.562753\pi$
$278$	0	0
$279$	−1.56681	−0.0938024
$280$	0	0
$281$	−2.61394	−0.155934	−0.0779672	−	0.996956i	$-0.524843\pi$
$281$	−2.61394	−0.155934	−0.0779672	+	0.996956i	$0.524843\pi$
$282$	0	0
$283$	9.65338	0.573834	0.286917	−	0.957955i	$-0.407370\pi$
$283$	9.65338	0.573834	0.286917	+	0.957955i	$0.407370\pi$
$284$	0	0
$285$	0.0254658	0.00150847
$286$	0	0
$287$	−11.4049	−0.673212
$288$	0	0
$289$	23.0666	1.35686
$290$	0	0
$291$	−8.38934	−0.491792
$292$	0	0
$293$	2.04724	0.119601	0.0598005	−	0.998210i	$-0.480954\pi$
$293$	2.04724	0.119601	0.0598005	+	0.998210i	$0.480954\pi$
$294$	0	0
$295$	−5.68809	−0.331173
$296$	0	0
$297$	20.0797	1.16514
$298$	0	0
$299$	3.01293	0.174242
$300$	0	0
$301$	5.87924	0.338874
$302$	0	0
$303$	−20.7100	−1.18976
$304$	0	0
$305$	2.79111	0.159818
$306$	0	0
$307$	18.6159	1.06246	0.531231	−	0.847227i	$-0.321729\pi$
$307$	18.6159	1.06246	0.531231	+	0.847227i	$0.321729\pi$
$308$	0	0
$309$	8.97956	0.510829
$310$	0	0
$311$	9.88503	0.560529	0.280264	−	0.959923i	$-0.409578\pi$
$311$	9.88503	0.560529	0.280264	+	0.959923i	$0.409578\pi$
$312$	0	0
$313$	17.4438	0.985983	0.492991	−	0.870034i	$-0.335903\pi$
$313$	17.4438	0.985983	0.492991	+	0.870034i	$0.335903\pi$
$314$	0	0
$315$	0.253565	0.0142868
$316$	0	0
$317$	2.92828	0.164468	0.0822342	−	0.996613i	$-0.473794\pi$
$317$	2.92828	0.164468	0.0822342	+	0.996613i	$0.473794\pi$
$318$	0	0
$319$	−7.66802	−0.429327
$320$	0	0
$321$	−5.70957	−0.318677
$322$	0	0
$323$	−0.134533	−0.00748563
$324$	0	0
$325$	−10.3292	−0.572963
$326$	0	0
$327$	11.2506	0.622159
$328$	0	0
$329$	5.01067	0.276247
$330$	0	0
$331$	−5.44140	−0.299087	−0.149543	−	0.988755i	$-0.547780\pi$
$331$	−5.44140	−0.299087	−0.149543	+	0.988755i	$0.547780\pi$
$332$	0	0
$333$	1.95145	0.106939
$334$	0	0
$335$	6.75407	0.369014
$336$	0	0
$337$	−3.73840	−0.203644	−0.101822	−	0.994803i	$-0.532467\pi$
$337$	−3.73840	−0.203644	−0.101822	+	0.994803i	$0.532467\pi$
$338$	0	0
$339$	−0.673551	−0.0365823
$340$	0	0
$341$	21.6439	1.17209
$342$	0	0
$343$	16.0010	0.863973
$344$	0	0
$345$	−1.56374	−0.0841892
$346$	0	0
$347$	−10.8089	−0.580252	−0.290126	−	0.956988i	$-0.593697\pi$
$347$	−10.8089	−0.580252	−0.290126	+	0.956988i	$0.593697\pi$
$348$	0	0
$349$	19.8978	1.06510	0.532552	−	0.846397i	$-0.321233\pi$
$349$	19.8978	1.06510	0.532552	+	0.846397i	$0.321233\pi$
$350$	0	0
$351$	12.4716	0.665683
$352$	0	0
$353$	−11.8505	−0.630736	−0.315368	−	0.948969i	$-0.602128\pi$
$353$	−11.8505	−0.630736	−0.315368	+	0.948969i	$0.602128\pi$
$354$	0	0
$355$	0.466273	0.0247472
$356$	0	0
$357$	13.5961	0.719584
$358$	0	0
$359$	−28.6632	−1.51279	−0.756393	−	0.654118i	$-0.773040\pi$
$359$	−28.6632	−1.51279	−0.756393	+	0.654118i	$0.773040\pi$
$360$	0	0
$361$	−18.9995	−0.999976
$362$	0	0
$363$	−4.65217	−0.244176
$364$	0	0
$365$	−2.16005	−0.113062
$366$	0	0
$367$	−13.1083	−0.684246	−0.342123	−	0.939655i	$-0.611146\pi$
$367$	−13.1083	−0.684246	−0.342123	+	0.939655i	$0.611146\pi$
$368$	0	0
$369$	−2.36092	−0.122905
$370$	0	0
$371$	8.92847	0.463543
$372$	0	0
$373$	−17.2738	−0.894406	−0.447203	−	0.894432i	$-0.647580\pi$
$373$	−17.2738	−0.894406	−0.447203	+	0.894432i	$0.647580\pi$
$374$	0	0
$375$	11.3519	0.586208
$376$	0	0
$377$	−4.76264	−0.245288
$378$	0	0
$379$	−6.30803	−0.324022	−0.162011	−	0.986789i	$-0.551798\pi$
$379$	−6.30803	−0.324022	−0.162011	+	0.986789i	$0.551798\pi$
$380$	0	0
$381$	−15.7421	−0.806490
$382$	0	0
$383$	28.2196	1.44196	0.720978	−	0.692958i	$-0.243693\pi$
$383$	28.2196	1.44196	0.720978	+	0.692958i	$0.243693\pi$
$384$	0	0
$385$	−3.50276	−0.178517
$386$	0	0
$387$	1.21705	0.0618663
$388$	0	0
$389$	−3.15029	−0.159726	−0.0798629	−	0.996806i	$-0.525448\pi$
$389$	−3.15029	−0.159726	−0.0798629	+	0.996806i	$0.525448\pi$
$390$	0	0
$391$	8.26110	0.417782
$392$	0	0
$393$	−36.2447	−1.82830
$394$	0	0
$395$	−0.303246	−0.0152579
$396$	0	0
$397$	31.2926	1.57053	0.785265	−	0.619159i	$-0.212527\pi$
$397$	31.2926	1.57053	0.785265	+	0.619159i	$0.212527\pi$
$398$	0	0
$399$	−0.0456523	−0.00228547
$400$	0	0
$401$	0.744072	0.0371572	0.0185786	−	0.999827i	$-0.494086\pi$
$401$	0.744072	0.0371572	0.0185786	+	0.999827i	$0.494086\pi$
$402$	0	0
$403$	13.4431	0.669651
$404$	0	0
$405$	−5.88764	−0.292559
$406$	0	0
$407$	−26.9574	−1.33623
$408$	0	0
$409$	3.60242	0.178128	0.0890641	−	0.996026i	$-0.471612\pi$
$409$	3.60242	0.178128	0.0890641	+	0.996026i	$0.471612\pi$
$410$	0	0
$411$	4.11237	0.202848
$412$	0	0
$413$	10.1970	0.501760
$414$	0	0
$415$	11.3340	0.556364
$416$	0	0
$417$	35.7355	1.74997
$418$	0	0
$419$	17.1323	0.836966	0.418483	−	0.908225i	$-0.362562\pi$
$419$	17.1323	0.836966	0.418483	+	0.908225i	$0.362562\pi$
$420$	0	0
$421$	24.1651	1.17773	0.588866	−	0.808230i	$-0.299574\pi$
$421$	24.1651	1.17773	0.588866	+	0.808230i	$0.299574\pi$
$422$	0	0
$423$	1.03725	0.0504329
$424$	0	0
$425$	−28.3216	−1.37380
$426$	0	0
$427$	−5.00359	−0.242141
$428$	0	0
$429$	−14.1800	−0.684615
$430$	0	0
$431$	−16.8823	−0.813193	−0.406596	−	0.913608i	$-0.633284\pi$
$431$	−16.8823	−0.813193	−0.406596	+	0.913608i	$0.633284\pi$
$432$	0	0
$433$	−13.0644	−0.627834	−0.313917	−	0.949450i	$-0.601641\pi$
$433$	−13.0644	−0.627834	−0.313917	+	0.949450i	$0.601641\pi$
$434$	0	0
$435$	2.47187	0.118517
$436$	0	0
$437$	−0.0277386	−0.00132692
$438$	0	0
$439$	−3.91599	−0.186900	−0.0934499	−	0.995624i	$-0.529789\pi$
$439$	−3.91599	−0.186900	−0.0934499	+	0.995624i	$0.529789\pi$
$440$	0	0
$441$	1.42889	0.0680425
$442$	0	0
$443$	−25.2673	−1.20049	−0.600243	−	0.799818i	$-0.704929\pi$
$443$	−25.2673	−1.20049	−0.600243	+	0.799818i	$0.704929\pi$
$444$	0	0
$445$	7.53165	0.357034
$446$	0	0
$447$	−34.5559	−1.63444
$448$	0	0
$449$	27.4706	1.29642	0.648210	−	0.761462i	$-0.275518\pi$
$449$	27.4706	1.29642	0.648210	+	0.761462i	$0.275518\pi$
$450$	0	0
$451$	32.6138	1.53573
$452$	0	0
$453$	−3.42423	−0.160885
$454$	0	0
$455$	−2.17558	−0.101993
$456$	0	0
$457$	−14.5345	−0.679896	−0.339948	−	0.940444i	$-0.610409\pi$
$457$	−14.5345	−0.679896	−0.339948	+	0.940444i	$0.610409\pi$
$458$	0	0
$459$	34.1956	1.59611
$460$	0	0
$461$	−10.4630	−0.487312	−0.243656	−	0.969862i	$-0.578347\pi$
$461$	−10.4630	−0.487312	−0.243656	+	0.969862i	$0.578347\pi$
$462$	0	0
$463$	27.7519	1.28974	0.644871	−	0.764292i	$-0.276911\pi$
$463$	27.7519	1.28974	0.644871	+	0.764292i	$0.276911\pi$
$464$	0	0
$465$	−6.97715	−0.323558
$466$	0	0
$467$	12.3322	0.570665	0.285333	−	0.958429i	$-0.407896\pi$
$467$	12.3322	0.570665	0.285333	+	0.958429i	$0.407896\pi$
$468$	0	0
$469$	−12.1080	−0.559093
$470$	0	0
$471$	4.31276	0.198721
$472$	0	0
$473$	−16.8124	−0.773036
$474$	0	0
$475$	0.0950965	0.00436333
$476$	0	0
$477$	1.84827	0.0846265
$478$	0	0
$479$	2.92366	0.133586	0.0667928	−	0.997767i	$-0.478723\pi$
$479$	2.92366	0.133586	0.0667928	+	0.997767i	$0.478723\pi$
$480$	0	0
$481$	−16.7434	−0.763432
$482$	0	0
$483$	2.80331	0.127555
$484$	0	0
$485$	3.68075	0.167134
$486$	0	0
$487$	−10.1739	−0.461025	−0.230513	−	0.973069i	$-0.574040\pi$
$487$	−10.1739	−0.461025	−0.230513	+	0.973069i	$0.574040\pi$
$488$	0	0
$489$	−10.9589	−0.495577
$490$	0	0
$491$	16.8181	0.758989	0.379494	−	0.925194i	$-0.376098\pi$
$491$	16.8181	0.758989	0.379494	+	0.925194i	$0.376098\pi$
$492$	0	0
$493$	−13.0586	−0.588130
$494$	0	0
$495$	−0.725101	−0.0325909
$496$	0	0
$497$	−0.835882	−0.0374944
$498$	0	0
$499$	30.0514	1.34529	0.672643	−	0.739967i	$-0.265159\pi$
$499$	30.0514	1.34529	0.672643	+	0.739967i	$0.265159\pi$
$500$	0	0
$501$	37.4088	1.67130
$502$	0	0
$503$	6.83648	0.304823	0.152412	−	0.988317i	$-0.451296\pi$
$503$	6.83648	0.304823	0.152412	+	0.988317i	$0.451296\pi$
$504$	0	0
$505$	9.08630	0.404335
$506$	0	0
$507$	12.6760	0.562960
$508$	0	0
$509$	−16.9547	−0.751502	−0.375751	−	0.926721i	$-0.622615\pi$
$509$	−16.9547	−0.751502	−0.375751	+	0.926721i	$0.622615\pi$
$510$	0	0
$511$	3.87230	0.171300
$512$	0	0
$513$	−0.114820	−0.00506943
$514$	0	0
$515$	−3.93970	−0.173604
$516$	0	0
$517$	−14.3286	−0.630172
$518$	0	0
$519$	−20.2366	−0.888287
$520$	0	0
$521$	37.1136	1.62597	0.812987	−	0.582282i	$-0.197840\pi$
$521$	37.1136	1.62597	0.812987	+	0.582282i	$0.197840\pi$
$522$	0	0
$523$	9.80481	0.428734	0.214367	−	0.976753i	$-0.431231\pi$
$523$	9.80481	0.428734	0.214367	+	0.976753i	$0.431231\pi$
$524$	0	0
$525$	−9.61060	−0.419441
$526$	0	0
$527$	36.8596	1.60563
$528$	0	0
$529$	−21.2967	−0.925943
$530$	0	0
$531$	2.11086	0.0916037
$532$	0	0
$533$	20.2566	0.877410
$534$	0	0
$535$	2.50502	0.108302
$536$	0	0
$537$	−29.7758	−1.28492
$538$	0	0
$539$	−19.7388	−0.850208
$540$	0	0
$541$	−15.4647	−0.664878	−0.332439	−	0.943125i	$-0.607871\pi$
$541$	−15.4647	−0.664878	−0.332439	+	0.943125i	$0.607871\pi$
$542$	0	0
$543$	−12.8517	−0.551521
$544$	0	0
$545$	−4.93610	−0.211439
$546$	0	0
$547$	31.5080	1.34719	0.673593	−	0.739103i	$-0.264750\pi$
$547$	31.5080	1.34719	0.673593	+	0.739103i	$0.264750\pi$
$548$	0	0
$549$	−1.03579	−0.0442063
$550$	0	0
$551$	0.0438474	0.00186796
$552$	0	0
$553$	0.543625	0.0231173
$554$	0	0
$555$	8.69001	0.368871
$556$	0	0
$557$	−5.45649	−0.231199	−0.115599	−	0.993296i	$-0.536879\pi$
$557$	−5.45649	−0.231199	−0.115599	+	0.993296i	$0.536879\pi$
$558$	0	0
$559$	−10.4423	−0.441660
$560$	0	0
$561$	−38.8798	−1.64151
$562$	0	0
$563$	5.91106	0.249121	0.124561	−	0.992212i	$-0.460248\pi$
$563$	5.91106	0.249121	0.124561	+	0.992212i	$0.460248\pi$
$564$	0	0
$565$	0.295514	0.0124324
$566$	0	0
$567$	10.5547	0.443256
$568$	0	0
$569$	31.9474	1.33930	0.669652	−	0.742675i	$-0.266443\pi$
$569$	31.9474	1.33930	0.669652	+	0.742675i	$0.266443\pi$
$570$	0	0
$571$	13.7554	0.575647	0.287824	−	0.957683i	$-0.407068\pi$
$571$	13.7554	0.575647	0.287824	+	0.957683i	$0.407068\pi$
$572$	0	0
$573$	0.515998	0.0215561
$574$	0	0
$575$	−5.83946	−0.243522
$576$	0	0
$577$	−2.93964	−0.122379	−0.0611893	−	0.998126i	$-0.519489\pi$
$577$	−2.93964	−0.122379	−0.0611893	+	0.998126i	$0.519489\pi$
$578$	0	0
$579$	0.556826	0.0231409
$580$	0	0
$581$	−20.3183	−0.842947
$582$	0	0
$583$	−25.5321	−1.05743
$584$	0	0
$585$	−0.450363	−0.0186202
$586$	0	0
$587$	−15.2876	−0.630989	−0.315494	−	0.948927i	$-0.602170\pi$
$587$	−15.2876	−0.630989	−0.315494	+	0.948927i	$0.602170\pi$
$588$	0	0
$589$	−0.123765	−0.00509964
$590$	0	0
$591$	−44.3340	−1.82366
$592$	0	0
$593$	−33.5005	−1.37570	−0.687850	−	0.725853i	$-0.741445\pi$
$593$	−33.5005	−1.37570	−0.687850	+	0.725853i	$0.741445\pi$
$594$	0	0
$595$	−5.96518	−0.244549
$596$	0	0
$597$	6.66904	0.272946
$598$	0	0
$599$	−0.150988	−0.00616920	−0.00308460	−	0.999995i	$-0.500982\pi$
$599$	−0.150988	−0.00616920	−0.00308460	+	0.999995i	$0.500982\pi$
$600$	0	0
$601$	−34.9546	−1.42583	−0.712915	−	0.701251i	$-0.752625\pi$
$601$	−34.9546	−1.42583	−0.712915	+	0.701251i	$0.752625\pi$
$602$	0	0
$603$	−2.50645	−0.102071
$604$	0	0
$605$	2.04110	0.0829824
$606$	0	0
$607$	−10.5301	−0.427404	−0.213702	−	0.976899i	$-0.568552\pi$
$607$	−10.5301	−0.427404	−0.213702	+	0.976899i	$0.568552\pi$
$608$	0	0
$609$	−4.43129	−0.179565
$610$	0	0
$611$	−8.89957	−0.360038
$612$	0	0
$613$	−20.2271	−0.816965	−0.408483	−	0.912766i	$-0.633942\pi$
$613$	−20.2271	−0.816965	−0.408483	+	0.912766i	$0.633942\pi$
$614$	0	0
$615$	−10.5134	−0.423942
$616$	0	0
$617$	7.82969	0.315211	0.157606	−	0.987502i	$-0.449622\pi$
$617$	7.82969	0.315211	0.157606	+	0.987502i	$0.449622\pi$
$618$	0	0
$619$	23.6992	0.952549	0.476275	−	0.879297i	$-0.341987\pi$
$619$	23.6992	0.952549	0.476275	+	0.879297i	$0.341987\pi$
$620$	0	0
$621$	7.05059	0.282931
$622$	0	0
$623$	−13.5019	−0.540942
$624$	0	0
$625$	17.3910	0.695642
$626$	0	0
$627$	0.130548	0.00521360
$628$	0	0
$629$	−45.9084	−1.83049
$630$	0	0
$631$	31.8687	1.26867	0.634336	−	0.773058i	$-0.281274\pi$
$631$	31.8687	1.26867	0.634336	+	0.773058i	$0.281274\pi$
$632$	0	0
$633$	15.2479	0.606050
$634$	0	0
$635$	6.90668	0.274083
$636$	0	0
$637$	−12.2598	−0.485752
$638$	0	0
$639$	−0.173035	−0.00684515
$640$	0	0
$641$	10.3578	0.409108	0.204554	−	0.978855i	$-0.434426\pi$
$641$	10.3578	0.409108	0.204554	+	0.978855i	$0.434426\pi$
$642$	0	0
$643$	21.9761	0.866653	0.433326	−	0.901237i	$-0.357340\pi$
$643$	21.9761	0.866653	0.433326	+	0.901237i	$0.357340\pi$
$644$	0	0
$645$	5.41966	0.213399
$646$	0	0
$647$	−9.10376	−0.357906	−0.178953	−	0.983858i	$-0.557271\pi$
$647$	−9.10376	−0.357906	−0.178953	+	0.983858i	$0.557271\pi$
$648$	0	0
$649$	−29.1595	−1.14461
$650$	0	0
$651$	12.5079	0.490222
$652$	0	0
$653$	6.84892	0.268019	0.134010	−	0.990980i	$-0.457215\pi$
$653$	6.84892	0.268019	0.134010	+	0.990980i	$0.457215\pi$
$654$	0	0
$655$	15.9020	0.621344
$656$	0	0
$657$	0.801599	0.0312734
$658$	0	0
$659$	10.9102	0.424999	0.212500	−	0.977161i	$-0.431840\pi$
$659$	10.9102	0.424999	0.212500	+	0.977161i	$0.431840\pi$
$660$	0	0
$661$	−9.26687	−0.360439	−0.180220	−	0.983626i	$-0.557681\pi$
$661$	−9.26687	−0.360439	−0.180220	+	0.983626i	$0.557681\pi$
$662$	0	0
$663$	−24.1484	−0.937847
$664$	0	0
$665$	0.0200295	0.000776712 0
$666$	0	0
$667$	−2.69248	−0.104253
$668$	0	0
$669$	−15.6140	−0.603673
$670$	0	0
$671$	14.3084	0.552369
$672$	0	0
$673$	−15.0272	−0.579255	−0.289627	−	0.957139i	$-0.593531\pi$
$673$	−15.0272	−0.579255	−0.289627	+	0.957139i	$0.593531\pi$
$674$	0	0
$675$	−24.1716	−0.930366
$676$	0	0
$677$	−46.3947	−1.78309	−0.891547	−	0.452929i	$-0.850379\pi$
$677$	−46.3947	−1.78309	−0.891547	+	0.452929i	$0.850379\pi$
$678$	0	0
$679$	−6.59843	−0.253225
$680$	0	0
$681$	25.1289	0.962941
$682$	0	0
$683$	34.2506	1.31056	0.655281	−	0.755385i	$-0.272550\pi$
$683$	34.2506	1.31056	0.655281	+	0.755385i	$0.272550\pi$
$684$	0	0
$685$	−1.80426	−0.0689373
$686$	0	0
$687$	−47.3310	−1.80579
$688$	0	0
$689$	−15.8581	−0.604144
$690$	0	0
$691$	1.19208	0.0453489	0.0226744	−	0.999743i	$-0.492782\pi$
$691$	1.19208	0.0453489	0.0226744	+	0.999743i	$0.492782\pi$
$692$	0	0
$693$	1.29988	0.0493784
$694$	0	0
$695$	−15.6786	−0.594723
$696$	0	0
$697$	55.5412	2.10377
$698$	0	0
$699$	−5.20348	−0.196814
$700$	0	0
$701$	−24.6698	−0.931767	−0.465883	−	0.884846i	$-0.654263\pi$
$701$	−24.6698	−0.931767	−0.465883	+	0.884846i	$0.654263\pi$
$702$	0	0
$703$	0.154149	0.00581382
$704$	0	0
$705$	4.61898	0.173961
$706$	0	0
$707$	−16.2889	−0.612608
$708$	0	0
$709$	23.3891	0.878398	0.439199	−	0.898390i	$-0.355262\pi$
$709$	23.3891	0.878398	0.439199	+	0.898390i	$0.355262\pi$
$710$	0	0
$711$	0.112535	0.00422040
$712$	0	0
$713$	7.59986	0.284617
$714$	0	0
$715$	6.22133	0.232665
$716$	0	0
$717$	20.6667	0.771812
$718$	0	0
$719$	1.52136	0.0567371	0.0283686	−	0.999598i	$-0.490969\pi$
$719$	1.52136	0.0567371	0.0283686	+	0.999598i	$0.490969\pi$
$720$	0	0
$721$	7.06265	0.263027
$722$	0	0
$723$	−6.16446	−0.229259
$724$	0	0
$725$	9.23065	0.342818
$726$	0	0
$727$	−36.3855	−1.34946	−0.674732	−	0.738063i	$-0.735741\pi$
$727$	−36.3855	−1.34946	−0.674732	+	0.738063i	$0.735741\pi$
$728$	0	0
$729$	28.9677	1.07288
$730$	0	0
$731$	−28.6315	−1.05897
$732$	0	0
$733$	−18.7869	−0.693908	−0.346954	−	0.937882i	$-0.612784\pi$
$733$	−18.7869	−0.693908	−0.346954	+	0.937882i	$0.612784\pi$
$734$	0	0
$735$	6.36299	0.234703
$736$	0	0
$737$	34.6242	1.27540
$738$	0	0
$739$	38.7776	1.42646	0.713228	−	0.700932i	$-0.247233\pi$
$739$	38.7776	1.42646	0.713228	+	0.700932i	$0.247233\pi$
$740$	0	0
$741$	0.0810841	0.00297870
$742$	0	0
$743$	6.80190	0.249537	0.124769	−	0.992186i	$-0.460181\pi$
$743$	6.80190	0.249537	0.124769	+	0.992186i	$0.460181\pi$
$744$	0	0
$745$	15.1611	0.555460
$746$	0	0
$747$	−4.20607	−0.153892
$748$	0	0
$749$	−4.49073	−0.164088
$750$	0	0
$751$	−10.5672	−0.385601	−0.192801	−	0.981238i	$-0.561757\pi$
$751$	−10.5672	−0.385601	−0.192801	+	0.981238i	$0.561757\pi$
$752$	0	0
$753$	−43.4283	−1.58262
$754$	0	0
$755$	1.50235	0.0546762
$756$	0	0
$757$	−22.6389	−0.822825	−0.411412	−	0.911449i	$-0.634964\pi$
$757$	−22.6389	−0.822825	−0.411412	+	0.911449i	$0.634964\pi$
$758$	0	0
$759$	−8.01641	−0.290977
$760$	0	0
$761$	1.83584	0.0665490	0.0332745	−	0.999446i	$-0.489406\pi$
$761$	1.83584	0.0665490	0.0332745	+	0.999446i	$0.489406\pi$
$762$	0	0
$763$	8.84889	0.320351
$764$	0	0
$765$	−1.23484	−0.0446459
$766$	0	0
$767$	−18.1111	−0.653954
$768$	0	0
$769$	50.0137	1.80354	0.901770	−	0.432215i	$-0.142268\pi$
$769$	50.0137	1.80354	0.901770	+	0.432215i	$0.142268\pi$
$770$	0	0
$771$	−46.6412	−1.67974
$772$	0	0
$773$	40.2988	1.44945	0.724724	−	0.689039i	$-0.241967\pi$
$773$	40.2988	1.44945	0.724724	+	0.689039i	$0.241967\pi$
$774$	0	0
$775$	−26.0546	−0.935911
$776$	0	0
$777$	−15.5785	−0.558875
$778$	0	0
$779$	−0.186493	−0.00668180
$780$	0	0
$781$	2.39031	0.0855319
$782$	0	0
$783$	−11.1451	−0.398294
$784$	0	0
$785$	−1.89218	−0.0675349
$786$	0	0
$787$	−38.2500	−1.36347	−0.681733	−	0.731601i	$-0.738773\pi$
$787$	−38.2500	−1.36347	−0.681733	+	0.731601i	$0.738773\pi$
$788$	0	0
$789$	41.9526	1.49355
$790$	0	0
$791$	−0.529765	−0.0188363
$792$	0	0
$793$	8.88700	0.315586
$794$	0	0
$795$	8.23053	0.291907
$796$	0	0
$797$	−33.4247	−1.18397	−0.591983	−	0.805951i	$-0.701655\pi$
$797$	−33.4247	−1.18397	−0.591983	+	0.805951i	$0.701655\pi$
$798$	0	0
$799$	−24.4016	−0.863266
$800$	0	0
$801$	−2.79501	−0.0987569
$802$	0	0
$803$	−11.0733	−0.390769
$804$	0	0
$805$	−1.22993	−0.0433492
$806$	0	0
$807$	1.65255	0.0581727
$808$	0	0
$809$	−39.8841	−1.40225	−0.701125	−	0.713038i	$-0.747319\pi$
$809$	−39.8841	−1.40225	−0.701125	+	0.713038i	$0.747319\pi$
$810$	0	0
$811$	−34.7237	−1.21931	−0.609657	−	0.792665i	$-0.708693\pi$
$811$	−34.7237	−1.21931	−0.609657	+	0.792665i	$0.708693\pi$
$812$	0	0
$813$	−29.5936	−1.03789
$814$	0	0
$815$	4.80810	0.168420
$816$	0	0
$817$	0.0961370	0.00336341
$818$	0	0
$819$	0.807361	0.0282115
$820$	0	0
$821$	−48.5169	−1.69325	−0.846626	−	0.532189i	$-0.821370\pi$
$821$	−48.5169	−1.69325	−0.846626	+	0.532189i	$0.821370\pi$
$822$	0	0
$823$	−19.5287	−0.680728	−0.340364	−	0.940294i	$-0.610550\pi$
$823$	−19.5287	−0.680728	−0.340364	+	0.940294i	$0.610550\pi$
$824$	0	0
$825$	27.4827	0.956825
$826$	0	0
$827$	−18.5400	−0.644698	−0.322349	−	0.946621i	$-0.604472\pi$
$827$	−18.5400	−0.644698	−0.322349	+	0.946621i	$0.604472\pi$
$828$	0	0
$829$	−20.2344	−0.702769	−0.351385	−	0.936231i	$-0.614289\pi$
$829$	−20.2344	−0.702769	−0.351385	+	0.936231i	$0.614289\pi$
$830$	0	0
$831$	10.7744	0.373762
$832$	0	0
$833$	−33.6150	−1.16469
$834$	0	0
$835$	−16.4128	−0.567987
$836$	0	0
$837$	31.4585	1.08737
$838$	0	0
$839$	−46.3931	−1.60167	−0.800833	−	0.598887i	$-0.795610\pi$
$839$	−46.3931	−1.60167	−0.800833	+	0.598887i	$0.795610\pi$
$840$	0	0
$841$	−24.7439	−0.853238
$842$	0	0
$843$	4.31967	0.148777
$844$	0	0
$845$	−5.56146	−0.191320
$846$	0	0
$847$	−3.65905	−0.125727
$848$	0	0
$849$	−15.9527	−0.547496
$850$	0	0
$851$	−9.46559	−0.324476
$852$	0	0
$853$	−0.133230	−0.00456172	−0.00228086	−	0.999997i	$-0.500726\pi$
$853$	−0.133230	−0.00456172	−0.00228086	+	0.999997i	$0.500726\pi$
$854$	0	0
$855$	0.00414629	0.000141800 0
$856$	0	0
$857$	−10.2942	−0.351642	−0.175821	−	0.984422i	$-0.556258\pi$
$857$	−10.2942	−0.351642	−0.175821	+	0.984422i	$0.556258\pi$
$858$	0	0
$859$	−35.5814	−1.21402	−0.607011	−	0.794694i	$-0.707632\pi$
$859$	−35.5814	−1.21402	−0.607011	+	0.794694i	$0.707632\pi$
$860$	0	0
$861$	18.8473	0.642313
$862$	0	0
$863$	−17.9896	−0.612372	−0.306186	−	0.951972i	$-0.599053\pi$
$863$	−17.9896	−0.612372	−0.306186	+	0.951972i	$0.599053\pi$
$864$	0	0
$865$	8.87861	0.301882
$866$	0	0
$867$	−38.1188	−1.29458
$868$	0	0
$869$	−1.55456	−0.0527350
$870$	0	0
$871$	21.5052	0.728677
$872$	0	0
$873$	−1.36593	−0.0462298
$874$	0	0
$875$	8.92853	0.301839
$876$	0	0
$877$	47.6131	1.60778	0.803890	−	0.594778i	$-0.202760\pi$
$877$	47.6131	1.60778	0.803890	+	0.594778i	$0.202760\pi$
$878$	0	0
$879$	−3.38318	−0.114112
$880$	0	0
$881$	−15.7287	−0.529914	−0.264957	−	0.964260i	$-0.585358\pi$
$881$	−15.7287	−0.529914	−0.264957	+	0.964260i	$0.585358\pi$
$882$	0	0
$883$	20.9111	0.703713	0.351856	−	0.936054i	$-0.385551\pi$
$883$	20.9111	0.703713	0.351856	+	0.936054i	$0.385551\pi$
$884$	0	0
$885$	9.39988	0.315973
$886$	0	0
$887$	−3.47543	−0.116694	−0.0583468	−	0.998296i	$-0.518583\pi$
$887$	−3.47543	−0.116694	−0.0583468	+	0.998296i	$0.518583\pi$
$888$	0	0
$889$	−12.3815	−0.415263
$890$	0	0
$891$	−30.1825	−1.01115
$892$	0	0
$893$	0.0819342	0.00274182
$894$	0	0
$895$	13.0639	0.436677
$896$	0	0
$897$	−4.97902	−0.166245
$898$	0	0
$899$	−12.0134	−0.400668
$900$	0	0
$901$	−43.4810	−1.44856
$902$	0	0
$903$	−9.71576	−0.323320
$904$	0	0
$905$	5.63859	0.187433
$906$	0	0
$907$	−5.99370	−0.199018	−0.0995088	−	0.995037i	$-0.531727\pi$
$907$	−5.99370	−0.199018	−0.0995088	+	0.995037i	$0.531727\pi$
$908$	0	0
$909$	−3.37195	−0.111840
$910$	0	0
$911$	−3.66404	−0.121395	−0.0606976	−	0.998156i	$-0.519333\pi$
$911$	−3.66404	−0.121395	−0.0606976	+	0.998156i	$0.519333\pi$
$912$	0	0
$913$	58.1028	1.92292
$914$	0	0
$915$	−4.61246	−0.152483
$916$	0	0
$917$	−28.5074	−0.941397
$918$	0	0
$919$	−44.8323	−1.47888	−0.739441	−	0.673222i	$-0.764910\pi$
$919$	−44.8323	−1.47888	−0.739441	+	0.673222i	$0.764910\pi$
$920$	0	0
$921$	−30.7637	−1.01370
$922$	0	0
$923$	1.48463	0.0488672
$924$	0	0
$925$	32.4510	1.06698
$926$	0	0
$927$	1.46203	0.0480194
$928$	0	0
$929$	−35.1823	−1.15429	−0.577146	−	0.816641i	$-0.695834\pi$
$929$	−35.1823	−1.15429	−0.577146	+	0.816641i	$0.695834\pi$
$930$	0	0
$931$	0.112870	0.00369918
$932$	0	0
$933$	−16.3355	−0.534802
$934$	0	0
$935$	17.0582	0.557862
$936$	0	0
$937$	37.7329	1.23268	0.616339	−	0.787481i	$-0.288615\pi$
$937$	37.7329	1.23268	0.616339	+	0.787481i	$0.288615\pi$
$938$	0	0
$939$	−28.8268	−0.940729
$940$	0	0
$941$	−31.0321	−1.01162	−0.505809	−	0.862646i	$-0.668806\pi$
$941$	−31.0321	−1.01162	−0.505809	+	0.862646i	$0.668806\pi$
$942$	0	0
$943$	11.4517	0.372919
$944$	0	0
$945$	−5.09110	−0.165613
$946$	0	0
$947$	−19.5388	−0.634924	−0.317462	−	0.948271i	$-0.602831\pi$
$947$	−19.5388	−0.634924	−0.317462	+	0.948271i	$0.602831\pi$
$948$	0	0
$949$	−6.87768	−0.223259
$950$	0	0
$951$	−4.83914	−0.156920
$952$	0	0
$953$	15.9894	0.517948	0.258974	−	0.965884i	$-0.416616\pi$
$953$	15.9894	0.517948	0.258974	+	0.965884i	$0.416616\pi$
$954$	0	0
$955$	−0.226389	−0.00732579
$956$	0	0
$957$	12.6718	0.409622
$958$	0	0
$959$	3.23448	0.104447
$960$	0	0
$961$	2.90921	0.0938456
$962$	0	0
$963$	−0.929619	−0.0299566
$964$	0	0
$965$	−0.244302	−0.00786437
$966$	0	0
$967$	−38.1787	−1.22774	−0.613872	−	0.789406i	$-0.710389\pi$
$967$	−38.1787	−1.22774	−0.613872	+	0.789406i	$0.710389\pi$
$968$	0	0
$969$	0.222323	0.00714206
$970$	0	0
$971$	−2.16411	−0.0694497	−0.0347248	−	0.999397i	$-0.511055\pi$
$971$	−2.16411	−0.0694497	−0.0347248	+	0.999397i	$0.511055\pi$
$972$	0	0
$973$	28.1069	0.901064
$974$	0	0
$975$	17.0696	0.546665
$976$	0	0
$977$	−14.7297	−0.471243	−0.235622	−	0.971845i	$-0.575713\pi$
$977$	−14.7297	−0.471243	−0.235622	+	0.971845i	$0.575713\pi$
$978$	0	0
$979$	38.6104	1.23399
$980$	0	0
$981$	1.83180	0.0584848
$982$	0	0
$983$	−36.4217	−1.16167	−0.580836	−	0.814020i	$-0.697274\pi$
$983$	−36.4217	−1.16167	−0.580836	+	0.814020i	$0.697274\pi$
$984$	0	0
$985$	19.4511	0.619764
$986$	0	0
$987$	−8.28040	−0.263568
$988$	0	0
$989$	−5.90336	−0.187716
$990$	0	0
$991$	24.5429	0.779632	0.389816	−	0.920893i	$-0.372539\pi$
$991$	24.5429	0.779632	0.389816	+	0.920893i	$0.372539\pi$
$992$	0	0
$993$	8.99221	0.285359
$994$	0	0
$995$	−2.92598	−0.0927598
$996$	0	0
$997$	−52.7413	−1.67033	−0.835167	−	0.549996i	$-0.814629\pi$
$997$	−52.7413	−1.67033	−0.835167	+	0.549996i	$0.814629\pi$
$998$	0	0
$999$	−39.1814	−1.23965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.h.1.3		7
4.3	odd	2		538.2.a.e.1.5	✓	7
12.11	even	2		4842.2.a.n.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.5	✓	7	4.3	odd	2
4304.2.a.h.1.3		7	1.1	even	1	trivial
4842.2.a.n.1.4		7	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65255 q^{3} +0.725043 q^{5} -1.29978 q^{7} -0.269065 q^{9} +3.71687 q^{11} +2.30856 q^{13} -1.19817 q^{15} +6.32982 q^{17} -0.0212539 q^{19} +2.14795 q^{21} +1.30511 q^{23} -4.47431 q^{25} +5.40231 q^{27} -2.06303 q^{29} +5.82316 q^{31} -6.14233 q^{33} -0.942394 q^{35} -7.25272 q^{37} -3.81503 q^{39} +8.77454 q^{41} -4.52327 q^{43} -0.195084 q^{45} -3.85502 q^{47} -5.31058 q^{49} -10.4604 q^{51} -6.86923 q^{53} +2.69489 q^{55} +0.0351232 q^{57} -7.84518 q^{59} +3.84958 q^{61} +0.349724 q^{63} +1.67381 q^{65} +9.31541 q^{67} -2.15676 q^{69} +0.643097 q^{71} -2.97920 q^{73} +7.39404 q^{75} -4.83110 q^{77} -0.418245 q^{79} -8.12041 q^{81} +15.6322 q^{83} +4.58939 q^{85} +3.40927 q^{87} +10.3879 q^{89} -3.00062 q^{91} -9.62309 q^{93} -0.0154100 q^{95} +5.07659 q^{97} -1.00008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} + 7 q^{5} - 6 q^{7} + 12 q^{9} + 3 q^{11} - 9 q^{13} - 8 q^{15} + 8 q^{17} + 11 q^{19} - 6 q^{21} - 12 q^{23} + 22 q^{25} + 14 q^{27} - 5 q^{29} - 14 q^{31} - 4 q^{33} + 4 q^{35} + 13 q^{37}+ \cdots + 37 q^{99}+O(q^{100}) \) 7 * q - q^3 + 7 * q^5 - 6 * q^7 + 12 * q^9 + 3 * q^11 - 9 * q^13 - 8 * q^15 + 8 * q^17 + 11 * q^19 - 6 * q^21 - 12 * q^23 + 22 * q^25 + 14 * q^27 - 5 * q^29 - 14 * q^31 - 4 * q^33 + 4 * q^35 + 13 * q^37 + 18 * q^39 + 12 * q^41 + 11 * q^43 + 3 * q^45 - 2 * q^47 + 15 * q^49 + 26 * q^51 + 19 * q^53 + 40 * q^55 - 12 * q^57 + 9 * q^59 - 3 * q^61 + 26 * q^63 - 10 * q^65 + 33 * q^67 - 64 * q^69 - 28 * q^71 - 14 * q^73 + 45 * q^75 + 10 * q^77 - 2 * q^79 + 15 * q^81 + 7 * q^83 - 16 * q^85 - 16 * q^87 + 18 * q^89 + 26 * q^91 + 6 * q^93 + 34 * q^95 - 4 * q^97 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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